Given:
- Radius of the circumcircle = 3 cm
- Altitude drawn to the hypotenuse = 2 cm

To find:
The area of the right-angled triangle.

Solution:

Step 1: Understanding the Properties of a Right-Angled Triangle
In a right-angled triangle, the circumcenter (the center of the circumcircle) coincides with the midpoint of the hypotenuse. Therefore, the radius of the circumcircle is half the length of the hypotenuse.

Step 2: Finding the Hypotenuse
Since the radius of the circumcircle is 3 cm, the hypotenuse of the right-angled triangle is 2 * 3 = 6 cm.

Step 3: Finding the Area of the Triangle
The area of a right-angled triangle can be calculated using the formula: Area = 1/2 * base * height.

In this case, the base is the altitude drawn to the hypotenuse, which is given as 2 cm.

Now, let's find the height of the triangle. To do this, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let the other two sides be a and b. We know that the hypotenuse is 6 cm (as found in step 2), and the altitude drawn to the hypotenuse is 2 cm.

Using the Pythagorean theorem, we have:
a^2 + b^2 = 6^2
a^2 + b^2 = 36

Since the altitude divides the triangle into two smaller triangles, we can consider one of them. Let's assume a as the base of this triangle and b as the height.

Using the given information, we have:
base = a = 2 cm
height = b

Using the Pythagorean theorem, we substitute the values:
2^2 + b^2 = 36
4 + b^2 = 36
b^2 = 36 - 4
b^2 = 32

Taking the square root of both sides, we get:
b = √32

Step 4: Calculating the Area
Now that we have the base (2 cm) and height (√32 cm), we can calculate the area of the triangle using the formula:
Area = 1/2 * base * height

Plugging in the values:
Area = 1/2 * 2 * √32
Area = √32

Since the question asks for the area in square centimeters, we need to simplify √32.

√32 = √(16 * 2) = √16 * √2 = 4√2

Therefore, the area of the right-angled triangle is 4√2 cm².

Step 5: Comparing the Answer Options
The given answer options are:
a) 6 cm²
b) 3 cm²
c) 4 cm²
d) 8 cm²

Comparing the calculated area (4√2 cm²) with the answer options, we can see that option 'A